Negative Binomial Regression model discrepancies

In a table, I've got total plant Counts (C), previous yr plant counts(PC), and multiple weather predictors(W; continuous data). The objective is examine the relationship between counts (C) and various weather (W) predictors, bearing in mind the counts (C) will depend on previous year counts (PC).

Examination of basic stats of (C) as well as various fitted regression models (Poisson, Zero inflated, etc) so far have all indicated negative binomial distribution fits my data best.

I am confused which model is most appropriate given my objective. I am also surprised that some result in the exact same output despite the model differences

1) log(C) = a + W

2) log(C) = a + W + PC

3) log(C) = a + W + (1*log(PC)) <- R will not accept log in my formula unless I enter 1* first

4) log(C) = a + W + offset(PC)

5) log(C) = a + W + offset(1*log(PC))

Q1) Which model is the one I want to use to address my objective? My instincts were that Model #3 was the correct model for me to use. But that model gives the same output as Model #1. I don't understand why that would happen.

Q2)Is it appropriate to use offset in this manner as Model #5? And how is using offset such as Model #5 any different from Model #3?

• Using an offset (model 5) says that you know the coefficient is unity. Model 3 estimates it. I am surprised you have to enter 1* so perhaps you need to show the precise code although then there is a danger that you will be told it is off-topic as a programming question – mdewey Jan 9 '17 at 11:32